Hilbert statistics of vorticity scaling in two-dimensional turbulence

Abstract

In this paper, the scaling property of the inverse energy cascade and forward enstrophy cascade of the vorticity filed ω(x, y) in two-dimensional (2D) turbulence is analyzed. This is accomplished by applying a Hilbert-based technique, namely Hilbert-Huang transform, to a vorticity field obtained from a 81922 grid-points direct numerical simulation of the 2D turbulence with a forcing scale kf = 100 and an Ekman friction. The measured joint probability density function p(C, k) of mode Ci(x) of the vorticity ω and instantaneous wavenumber k(x) is separated by the forcing scale kf into two parts, which correspond to the inverse energy cascade and the forward enstrophy cascade. It is found that all conditional probability density function p(C|k) at given wavenumber k has an exponential tail. In the inverse energy cascade, the shape of p(C|k) does collapse with each other, indicating a nonintermittent cascade. The measured scaling exponent \documentclass[12pt]{minimal}\begin{document}$\zeta _{\omega }^I(q)$\end{document}ζωI(q) is linear with the statistical order q, i.e., \documentclass[12pt]{minimal}\begin{document}$\zeta _{\omega }^I(q)=-q/3$\end{document}ζωI(q)=−q/3, confirming the nonintermittent cascade process. In the forward enstrophy cascade, the core part of p(C|k) is changing with wavenumber k, indicating an intermittent forward cascade. The measured scaling exponent \documentclass[12pt]{minimal}\begin{document}$\zeta _{\omega }^F(q)$\end{document}ζωF(q) is nonlinear with q and can be described very well by a log-Poisson fitting: \documentclass[12pt]{minimal}\begin{document}$\zeta _{\omega }^F(q)=\frac{1}{3}q+0.45\left( 1-0.43^{q}\right)$\end{document}ζωF(q)=13q+0.451−0.43q. However, the extracted vorticity scaling exponents ζω(q) for both inverse energy cascade and forward enstrophy cascade are not consistent with Kraichnan's theory prediction. New theory for the vorticity field in 2D turbulence is required to interpret the observed scaling behavior.

Multi-level segment analysis: definition and application in turbulent systems

“…Theoretically, Nam et al (2000) found that the Ekman friction leads to an intermittent forward enstrophy cascade (Bernard 2000), which has been verified indirectly by studying the passive scalar field, instead of the vorticity field (Boffetta et al 2002). More recently, this claim has been confirmed by Tan et al (2014) using Hilbert spectral analysis. A log-Poisson model without justice is proposed to fit the forward enstrophy cascade scaling exponent, see details in Tan et al (2014).…”

“…For the last few decades, numerous experiments and numerical simulations have been devoted to verifying the abovementioned forward and inverse cascades (Kellay et al 1998(Kellay et al , doi:10.1088(Kellay et al /1742(Kellay et al -5468/2015/06/P06018 Rutgers 1998, Bruneau and Kellay 2005, Bernard et al 2006, Boffetta 2007, Falkovich and Lebedev 2011, Tan et al 2014 with a partial verification of the theory by Kraichnan. For example, Boffetta and Musacchio (2010) performed a very high resolution numerical simulation, up to a grid number N = 32 768 2 .…”

“…The inverse and forward cascades were observed for the third-order velocity structure function as predicted by the theory (Boffetta and Musacchio 2010, see figure 3). For the inverse cascade, it is found that the inverse energy cascade is almost non-intermittent (Nam et al 2000, Tan et al 2014, while for the forward enstrophy cascade, the intermittency effect still remains an open question. This is because, according to the convergency condition, the structure-function requires a Fourier spectrum E(k) ∼ k −β with β ∈ (1, 3), the β-limitation (Huang et al 2010).…”

Two-scale correlation and energy cascade in three-dimensional turbulent flows

“…in which r N (t) is a residual. The extracted C i (t) can be considered as an 'intrinsic' oscillation structure of the turbulence, e.g., a vortex in the statistics sense [45]. Based on the dyadic filter bank property of the EMD methodology, the number N of the IMF mode is limited as…”

“…Lagrangian turbulence [24], river turbulence [29], vorticity statistics in two-dimensional turbulence [45], etc, to characterize the intermittent nature of these processes [23]. For details on this methodology, we refer to [22,21,18,28,23,27,26].…”